The limit. The dominant n-bonacci root ηₙ climbs monotonically toward 2: φ ≈ 1.618, η ≈ 1.839, Δ ≈ 1.928, Σ ≈ 1.966, Ω ≈ 1.984 … → τ = 2 as n → ∞. This is no_return_to_critical reframed — once the ladder passes a rung it never falls back below it.
The DNLS counterpart (Version 4). The same monotone climb has a dynamical face. Back in dm³ 102 (Week 6 · the DNLS lab) you saw tribonacci resist nonlinearity better than Fibonacci at λ = 1.5. Version 4 of the study asks what that resistance becomes at the extremes, and finds not a bigger number but a different regime.
Self-trapping threshold gap. Sweeping λ ∈ {1, 2, 4, 8, 10} at N=500, T=10⁵: the Fibonacci chain crosses into the self-trapped regime (spreading rate α ≈ 0) somewhere in λ ∈ [4, 8]. The tribonacci chain has not self-trapped by λ = 10 (α ≥ 0.09 throughout). Because η > φ, the tribonacci threshold sits higher — a difference in kind, not magnitude.
Finite-size scaling. Across N ∈ {500, 1000, 2000} at T=10⁴, the ratio IPRtrib/IPRfib generally grows with N — the differential persists toward the thermodynamic limit.
Long-time honesty. Run to T=10⁶ (N=1000, λ=1.5), the ratio peaks at 4.18× near T=10⁴, then both chains saturate to chain-limited IPR ≈ 0.002. The peak is a transient finite-size feature — reported, not hidden.
The bridge. The spectral-gap-controls-threshold mechanism (higher rung → higher self-trapping threshold, because ηₙ climbs toward 2) is the nonlinear-dynamics counterpart of the criticality line λc(n) ≈ 0.958 Δₙ + 0.107 across n = 2…5 (r = 0.989) in the companion criticality paper. Full V4: 10.5281/zenodo.20230642 · companion site.
1. From grossi-ops/Atratores, run dnls_long_time.py at N=500, T=10⁵ for λ ∈ {1,2,4,8,10}. Locate the Fibonacci self-trapping window. Confirm tribonacci stays untrapped through λ=10.
2. Add a hexanacci (n=6) chain. Predict, then measure: is its threshold higher than tribonacci's? Does λc(6) fall on the line λc(n) ≈ 0.958 Δₙ + 0.107? This closes an open regression point.
3. D2 writing task: state the regime distinction as a fixed-point argument. The self-trapped state is a fixed point of the DNLS flow; argue why raising the rung raises the λ at which that fixed point becomes attracting.
-- dm³ 103 · Week 06 · Lean 4 Lab · the monotone climb -- ηₙ = dominant real root of xⁿ = xⁿ⁻¹ + … + 1 ; ηₙ ↑ 2 theorem nbonacci_root_lt_two (n : ℕ) (hn : 2 ≤ n) : nbonacciRoot n < 2 := by sorry -- ← exercise: the ladder never reaches τ at finite n (no_return_to_critical) -- OPEN on the AXLE roadmap — do NOT mark done: -- IPR_trib(0) > IPR_fib(0) (the linear-limit inequality) -- Lean statement of the self-trapping threshold gap -- These are the frontier. Naming them honestly is the D2 deliverable.